Coupling time evolution model with empirical regression model to estimate mechanical wear

ABSTRACT

Mechanical systems wear or change over time. Data collected over a system&#39;s life can be input to statistical learning models to predict this wear/change. Previous work by the inventors trained a flexible empirical regression model at a fixed point of wear, and then applied it independently at time points over the life of an engine to predict wear. The embodiment disclosed herein relates those wear predictions over time using a time evolution model. The time evolution model is sequentially updated with new data, and effectively tunes the empirical model for each engine. The combined model predicts wear with dramatically reduced variability. The benefit of reduced variability is that engine wear is more evident, and it is possible to detect operational anomalies more quickly. In addition to tracking wear, the model is also used as the basis for a Bayesian approach to monitor for sudden changes and reject outliers, and adapt the model after these events.

BACKGROUND

This disclosure relates to a data-driven method and system forestimating and tracking accurate operational states of mechanicalsystems. In particular, the disclosure relates to a data-driven methodand system for estimating and tracking operational states, such as wearor anomalies over time, of mechanical systems.

Mechanical systems, such as engines, turbines, tires, brakes, and othersystem components, found in aircraft, automobiles, trucks, watercraft,power generator units, military vehicles, and other vehicles, wear orchange over time. Wear affects the performance of such mechanicalsystems. A key factor in monitoring the health of a mechanical system isto measure system wear as it occurs over time. Such monitoring can aidin maintenance planning and timely repair or replacement of themechanical system or components thereof. For example, with gas turbineengines, to get the same thrust output as an engine wears, the enginerequires more fuel, and the engine's exhaust gas temperature (EGT), asit leaves the engine, increases. However, EGT is also affected byoutside variables, such as environmental influences (e.g., temperatureand air quality), flight conditions, system faults, and other engineparameters for any given flight or data point. Such factors mayoverwhelm the EGT value more than wear for a given data point.Typically, engine wear is not evident in a time series plot of raw EGTdata plotted over the lifetime of an engine. Thus, EGT by itself may notreveal engine wear that is hidden by the variability due toenvironmental, operational and other factors.

Known methods and systems exist for monitoring and estimating the wearof a mechanical system. Empirical methods and systems for estimatingwear typically manually manipulate recorded data into tables for lookup.Such manual empirical methods are limited in the amount of data that canbe assembled and are not in an automated format to create a predictionmodel. In addition, such manual methods may be imprecise because of theoutside influences discussed above. Other known methods and systems usetheoretical models of the mechanical system which use physics orengineering information to build a model using test data. Such modelingis based on understanding how a system operates and progresses to afailure via knowledge, for example, of material properties and responseto loading. However, such physics or engineering model-based methods mayuse simplifying assumptions and are theoretical in nature. Moreover,such methods and systems only collect data when the engine is new and donot continually collect data during flights or track degradation of asystem over time. Outside influences, as in the empirical method, maynot be accounted for.

With regard to known methods and systems that estimate mechanical systemwear over time, the resulting output might be plotted over time toobserve trends. The scatter of the individual points is large enoughthat large rolling averages are required to obtain a value that can beused with confidence. This can cause time delays for any correctiveaction that may be needed and also for prediction of scheduledmaintenance for the engine.

Data collected over a system's life can be input to statistical learningmodels to estimate and track wear/change in a mechanical system. U.S.Patent Application Publ. No. 2010/0082267 (incorporated by referenceherein) discloses an automated data-driven method for estimating one ormore operational states, such as wear or degradation, of a mechanicalsystem over time. The method comprises training a regression model at afixed point of wear, and then applying it independently at time pointsover the life of the system to estimate wear. More specifically, themethod comprises the steps of collecting data on the mechanical systemfrom a data recording device, preprocessing the collected data,selecting a training data set that represents a base condition forstatistical comparison, fitting a statistical model to the training dataset to relate a system output to variables at the base condition, andusing an output model to predict what an observed response would havebeen at the base condition and calculating the difference between theobserved response and the predicted response to estimate the one or moreoperational states of the mechanical system. In particular, U.S. PatentApplication Publ. No. 2010/0082267 discloses a procedure thatempirically relates mechanical system output (e.g., engine EGT) to otherfactors (e.g., environmental, flight and mechanical parameters). Theresiduals are the difference between the observed mechanical systemoutput (e.g., engine EGT) and the output predicted by the model, andrepresent mechanical system wear over time or operational anomaly (partfailure).

As a baseline for comparison, an aircraft owner is typically providednormalized EGT data schedules by the engine manufacturer (for brevity,“OEM”). A previous investigation described by Basu et al. [see“Statistical Methods for Modeling and Predicting Maximum Engine ExhaustGas Temperature (EGT): First Analysis Using Climb Data from a SingleAircraft”, Networked Systems Technology Technical Report (NST-08-001)(2008) and “Regression Based Method for Predicting Engine Wear fromExhaust Gas Temperature”, Prognostics and Health Management Conference,Denver, Colo. (2008)] showed that a data-driven approach outperformedthe OEM results in the sense that its predictions (using a randomforest) had a similar range for estimating engine wear, but about 25%smaller variation.

There is a need for a data-driven method and system that further reducesvariability in estimation for operational states such as wear, and alsomonitors for more abrupt changes in the condition of mechanical systems.

SUMMARY

Data collected over a system's life can be input to statistical learningmodels to focus data model results for estimating wear/change ofmechanical systems. The wear of a mechanical system can be accuratelytracked using (1) data collected during use and (2) data-drivenstatistical models. In the particular example of a gas turbine aircraftengine, exhaust gas temperature (EGT) can be modeled as a function ofother recorded parameters. However, application of the techniquesdisclosed herein is not limited to gas turbines. These techniques couldalso be used for other systems that slowly change (degrade) over time.

The embodiment disclosed in detail hereinafter adopts a data-drivenapproach to reduce the variability of normalized EGT by accounting forboth (1) the effect of other variables and (2) time dependence. Theapproach builds a data-driven model using volumes of flight data thatare increasingly collected routinely on modern aircraft. Such adata-driven approach contrasts with a physics model approach developedusing physics/engineering insight and test data.

In the previous work by Basu et al. mentioned above, a flexibleempirical regression model was trained at a fixed point of wear, andthen applied independently at time points over the life of an engine toestimate wear. However, wear typically occurs slowly and smoothly. Inaccordance with the embodiment disclosed herein, a dynamic linear model(which is an example of a state space method) is coupled with theempirical regression model to provide the benefit of relating wearestimations over time. The combined model estimates operational states,such as wear or anomalies over time, of mechanical systems with reducedvariability in the estimations as compared to the empirical regressionmodel alone and a baseline method. The benefit of reduced variability isthat mechanical wear is more evident, and it is possible to detectoperational anomalies more quickly.

Coupling a time evolution model (e.g., a dynamic linear model) with anempirical regression model (e.g., random forest) reduces variability byaccounting for time dependence. The inventors have conducted severalexperiments that show that using DLMs to relate estimations over timedramatically improves the previously developed data-driven approachdescribed by Basu et al., which itself improves upon the OEM approach.

In addition to tracking wear, the combined model is also used as thebasis for a Bayesian approach to monitor for sudden changes and rejectoutliers, and adapt the model after these events. The monitor can beutilized for fault detection and prognosis.

By accurately tracking wear, the embodiment disclosed herein can quicklyobserve sharp failures and more quickly spot trend outside of normalbehavior. Since the methodology is data driven, and based on empiricalmodels that can be applied to many subsystems, it potentially savescosts over a detailed and expensive analysis based onphysics/engineering principles that needs to be conducted for each newlyencountered subsystem.

Similar normalization problems occur in other contexts. One example isestimating tire pressure loss. The ultimate goal may be to monitor wearor degradation as equipment is used, in order to repair or replace theequipment in a timely manner. The common elements of the approach tosuch problems disclosed herein are to (1) adjust a quantity of interestfor other influences, and (2) relate these adjustments over time.

The above-described subject matter may also be implemented in variousother embodiments without departing from the scope of the appendedclaims. These and various other features will be apparent from a readingof the Detailed Description with reference to the associated drawings.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a block flow diagram showing one embodiment of the data drivensystem and method disclosed in U.S. Patent Application Publ. No.2010/0082267.

FIGS. 2 and 3 are block flow diagrams showing the methodology combiningempirical regression and dynamic linear models in accordance with oneembodiment of the invention.

FIGS. 4A and 4B are block flow diagrams which, when viewed inconjunction, show a system in accordance with one embodiment of theinvention.

Reference will hereinafter be made to the drawings in which similarelements in different drawings bear the same reference numerals.

DETAILED DESCRIPTION

The method and system of the disclosed embodiment may be used with anymechanical system that produces data that can be analyzed, such asengines, turbines, oil systems, water cooling systems, or fornon-traditional systems that have periodic observations, such as tiresand brakes found in aircraft, automobiles and trucks, military vehicles,and other vehicles. Accordingly, one of ordinary skill in the art willrecognize and appreciate that the method and system disclosed herein canbe used in any number of applications involving the estimating ormonitoring of one or more operational states of a mechanical system overtime.

One goal of any integrated vehicle health management program foraircraft is to monitor its engine's health, in particular, engineperformance. To get the same thrust output as the engine wears, theengine requires more fuel, and so the engine's exhaust gas temperature(EGT) increases. However, environmental, flight, and other engineparameters also affect EGT.

To aid in understanding what follows, some technical background onadaptive nonparametric regression and state space methodology will bebriefly discussed. Then the rationale for fitting a regression model todata collected at baseline conditions, producing residuals, and relatingthese over time using a time evolution model will be explained. Lastly amonitor will be disclosed that employs a Bayes factor (similar tolikelihood ratio) to detect outliers, monitor for sudden shifts, andautomatically reject outliers and adapt the time evolution model tochanges.

TECHNICAL BACKGROUND

Focusing the discussion on regression, a variety of strategies relate aresponse variable y to a set of covariates x. The classical approach isto postulate a parametric function to predict y at the point x.Harnessing increasingly powerful yet cheap computer resources is onealternative to assuming a parametric model.

Tree-based methods are an example of adaptive nonparametric statisticalprocedures. Trees can capture non-linear relationships and interactionsamong predictors. The idea of tree-based regression is to partitioncovariate space into regions with homogeneous response variables. Arecursive partitioning algorithm starts by using a splitting rule todivide the training data into two groups. This procedure is recursivelyapplied to each group until the final groups contain only a fewobservations. These terminal nodes form a partition of the covariatespace which is conveniently represented as a binary tree.

Trees score high in interpretability, but not as high in prediction. Toimprove prediction, a variety of techniques (e.g., bagging, boosting,random forests) grow an ensemble of trees, each fit to a perturbedversion of the training set. These procedures are motivated by theobservation that slight changes in the data can lead to different treestructures, but comparable error rates. Fitting trees to deliberatelyperturbed training data produces a set of plausible models, eachachieved by the greedy algorithm converging to different local maxima.Rather than choosing one best model and discarding the rest, theresulting set of plausible models are combined to achieve superioraccuracy.

The common method of perturbing data is to bootstrap, i.e., sample withreplacement from the original data set. Bagging stands for BootstrapAggregation. A tree model is fit to each of several bootstrappedsamples. In regression, predictions are obtained by averaging thepredictions over the trees. Just as an average has lower variance than asingle measurement, bagging reduces variance. This is especiallyeffective when bagging unstable predictors like trees.

As with bagging, random forests use the bootstrap to perturb the data.In addition, they introduce another random element into the treeconstruction. At each node, a random set of predictor variables ischosen. The best split for the node is found by searching only over thisset, and not over all predictor variables. This additional randomnessallows variables to occur in the tree model that would not otherwiseappear in greedy search approaches, and often helps achieve greateraccuracy. The random forests algorithm is disclosed by Breiman, L.(2001), “Random Forests”, Machine Learning, Volume 45, Number 1, pages5-32.

State space models provide a flexible yet relatively simple tool foranalyzing dynamic phenomena and evolving systems, and extend classicalstatistical analysis to non-stationary processes. Informally, a statespace model consists of: (1) unobserved state variables whose dynamicsare described by a Markov dependency; and (2) observations, which areindependent conditional on the state variables. They allow interpretinga time series as the combination of several components, such as trend,seasonality, or regression. State estimation and forecasting are solvedby recursively computing the conditional distribution of the quantitiesof interest, given the available information, and hence can naturally betreated within a Bayesian framework.

The goal of recursive Bayesian estimation is to estimate an unknownprobability density function over time using observations and amathematical process model. A Bayes filter uses information about noiseand system dynamics to reduce uncertainty from noisy observations. Therecursive algorithm consists of two steps at each time: predict andupdate, which involve state transition and observation equations. Thepredict step uses the state estimate from the previous time to producean a priori state estimate at the current time, which is then updated bycombining with current observation information to produce an aposteriori state estimate.

The specific model used in this work is an example of a Dynamic LinearModel (DLM), which is a state space model that is linear and Gaussian.Such modeling is described by West and Harrison in “Bayesian Forecastingand Dynamic Models,” Springer (1999) and by Pole et al. in “AppliedBayesian Forecasting and Time Series Analysis,” Chapman & Hall (1994)(hereinafter “Pole et al. (1994)”). Dynamic Linear Models offer aflexible framework to model both smooth and abrupt changes in timeseries, and accommodate subjective information. In its simplest form, aDLM is estimated using a Kalman filter, although Markov Chain MonteCarlo (MCMC) and particle filtering are used for more complex versions,as disclosed by Petris et al. in “Dynamic Linear Models with R”,Springer (2009) (hereinafter “Petris et al. (2009)”).

The observation equation of a DLM is a multivariate regression thatrelates the observation to trend, regressors, seasonality and othercomponents. A system equation describes the evolution of the regressioncoefficients (state parameters) through time.

Prior Data-Driven Approach

U.S. Patent Application Publ. No. 2010/0082267 discloses a method fortraining a flexible empirical regression model at a fixed point of wear,and then applying it independently at time points over the life of anengine to estimate wear. The method estimates wear as the residuals thatresult from using statistical regression.

The method in accordance with one embodiment comprises the step ofcollecting data on the mechanical system via a data collectioncomponent, such as from a data recording device. The data collected mayinclude numerous technical parameters specific to a particularmechanical system, as well as various environmental and operationaldata. For example, the data collected for an aircraft engine mayinclude, but is not limited to, such data as exhaust gas temperature(EGT), Mach number, engine spool speeds, pressure altitude, total airtemperature, calibrated air speed, oil pressure, oil quantity, sensorinformation, fuel flow actual and commanded, engine pressure ratioactual and commanded, flight mode, throttle lever angle, and othersuitable data. In addition, the data recording device for collectingdata for an aircraft engine may be a Quick Access Recorder (QAR) on anaircraft. Such data may be recorded and collected from a single aircraftor multiple aircraft over the lifetime of the mechanical system. Thedata may be recorded and collected during initial climb of the aircraftover many flights, during descent of the aircraft over many flights,during cruise of the aircraft over many flights, or during otheroperational modes of the aircraft. The collecting data step may furthercomprise the step of determining one or more technical parameters of themechanical system to be measured relevant to the estimating of the oneor more operational states. Such technical parameters may be determinedor measured via the use of one or more sensors on the mechanical system.

The method further comprises the step of preprocessing the collectedmechanical system data to summarize operation of the mechanical system.The preprocessing step transforms the large amounts of mechanical systemdata collected into snapshots of data representative of the health ofthe mechanical system that a user is most interested in. Thepreprocessing step determines when during the operation of themechanical system the data is collected, what parameters of themechanical system are to be monitored or estimated, and how to reducethe amount of data to determine the results. The preprocessing softwarecode may be written in any suitable software programming environment. Byexample, with aircraft, the data (e.g., propulsion flight data) may besummarized from a single aircraft flight or multiple aircraft flightsover the lifetime of the mechanical system. In order to trend aircraftengine EGT over time, the most stable engine operational point possiblemust be obtained for each flight, ideally at or near the maximum engineoperating point. During flight this occurs in two modes: cruise andtakeoff. Very stable engine data and aircraft parametric data may berecorded during aircraft cruise. However, the aircraft may not reachcruise during every flight (e.g. during training). Also, during cruise,the engine is not near its maximum operation. During takeoff, the engineis operating near or at its maximum for a fixed period of time, butaircraft parametric and other engine data may be changing. The algorithmof the method finds a window of data in which maximum EGT occurs.

As described above, the method comprises the step of selecting atraining data set that represents a base condition for statisticalcomparison. The training data set is selected using subject matterexpert (SME) input. This step creates training data sets during periodsof time when the mechanical system output (e.g., engine EGT) isrelatively stable, yet there is sufficient variability in other outsidefactors (e.g., environmental, flight, and mechanical parameters) torepresent the variety of conditions under which the mechanical system(e.g., aircraft engine) operates. This step creates appropriate trainingdata sets of the mechanical system's environment (e.g., aircraft flightand engine variables), modeled from any source as long as it is from aconsistent wear position for the system. In other words, it is importantto ensure that the trained model represents a consistent reference orbase point condition in the life of the system. For example, withengines, training data is preferably selected from the middle of theengine's life when engine wear is relatively slow and stable. If thereis no low wear state in the life of other systems, such as tire wear,then using data from the time when the system is new is sufficient. Ifselected correctly, this training data will contain flights thatrepresent a typical set of environmental, flight, and engine parameterconditions that are diverse enough to represent the conditionsencountered in future operations. Selecting training data in this wayimproves results over using data from the entire engine lifetime.However, performance may decrease if the slice of data is too small,leading to insufficient covariate diversity.

The method further comprises the step of fitting a statistical model tothe training data set to relate a mechanical system output to variablesat the base condition. The statistical model may comprise linearregression, non-linear regression, or adaptive nonparametric proceduressuch as random forest techniques. A random forest is an example of atree ensemble which is a nonparametric statistical technique. Anonparametric random forest technique or model may be preferred becauseit is flexible and is data driven and one does not need to specify whatthe relationship should be. This step empirically relates the mechanicalsystem output (e.g., engine EGT) or predicted response to outsidevariables, such as environmental influences (e.g., temperature and airquality), flight conditions, system faults, and mechanical parameters.The modeling software code may be written in any suitable softwareprogramming environment. For new environmental (e.g., flight and engine)data, the model predicts the mechanical system output (e.g., EGT) usingthe trained model.

The method further comprises the step of using an output model topredict what an observed response would have been at the base conditionand calculating the difference between the observed response and thepredicted response to estimate the one or more operational states of themechanical system. The predicting component uses the output model topredict the operational states of the mechanical system. The predictingcomponent may be in the form of suitable modeling software. This stepgenerates or calculates the estimated operational states as theresiduals, which are the difference between the observed mechanicalsystem output (e.g., EGT) (part of the preprocessed data) and thepredicted response. The residuals may represent mechanical system wearover time (e.g., engine performance degradation as noted by mechanicalenergy needed to produce the same work) or operational anomaly (partfailure). The flexible statistical models can be applied to datacollected over the mechanical system's life, to account for irrelevantor nuisance factors, and to generate a predictive model of operationalstates such as wear, degradation or anomalies. Using the model adjustsfor environmental and other outside factors and references back to theconsistent wear or base point condition. By example, for aircraftengines, the method may use the statistical model to model aircraft gasturbine engine EGT as a function of environmental, flight and engineparameters and to generate the residuals as an estimation of enginewear. The method may further comprise the step of using the estimatedone or more operational states for trend analysis.

FIG. 1 is taken from U.S. Patent Application Publ. No. 2010/0082267 andshows an automated data-driven method 100 for estimating wear of amechanical system over time as described in the preceding paragraphs.The method comprises the step 102 of collecting data on the mechanicalsystem from a data recording device. The result of step 102 is rawrecorded mechanical system data 104. The collecting data step mayfurther comprise the step of determining one or more technicalparameters of the mechanical system to be measured relevant to theestimating of the one or more operational states. The method shown inFIG. 1 further comprises the step 106 of determining and measuring oneor more technical parameters of the mechanical system relevant orimportant to the estimating of the one or more operational states. SME(subject matter expert) input 108 may be used to input various technicalparameters of a particular mechanical system in order to determine andmeasure such technical parameters. The result of step 106 is relevantmechanical system data 110. Method 100 further comprises the step 112 ofpreprocessing the collected mechanical system data to summarizeoperation of the mechanical system, as previously described. Thepreprocessing software code may be written in Statistical AnalysisSoftware (SAS) or another suitable commercially available software code.The result of step 112 is preprocessed mechanical system data 114.

A determination is then made in step 116 whether a wear model exists. Ifa wear model exists, then the steps following the arrow labeled “YES” inFIG. 1 are followed. If a wear model does not exist, then the stepsfollowing the arrow labeled “NO” in FIG. 1 are followed. In response toa determination that a wear model does not exist, a SME performs thestep 118 of selecting a consistent or reference wear data point (basecondition) of the mechanical system. This step includes selecting atraining data set that represents a base condition for statisticalcomparison. In order to build a statistical model, it is preferable tohave a consistent known starting point. Subject Matter Expert input 120may be used to select the reference wear data point. The result of step118 is a training data subset 122.

As shown in FIG. 1, method 100 further comprises the step 124 of fittinga statistical model to the training data set to relate a predictedresponse to outside variables at the base condition, that is, that thewear data point is a base point for statistical comparison. Such outsidevariables can include environmental factors (e.g., temperature and airquality), flight information, system faults, and mechanical parameters.This step trains a statistical model at the reference wear data point asthe basis for statistical comparison and includes determining whichtechnical parameters are important for estimation or monitoring. Theresulting output model 126 is then used with the preprocessed systemdata in step 128 to predict what an observed response would have been atthe base condition and calculating the difference between the observedresponse and the predicted response to estimate the wear of themechanical system. As previously discussed, step 128 generates orcalculates the estimated operational states or residuals, which is thedifference between the observed response or mechanical system output(e.g., EGT) and the predicted response. The result of step 128 isprocessed data 130 that represents wear.

For new environmental (e.g., flight and engine) data, the model predictsthe mechanical system output (e.g., EGT) using the trained model. Theresiduals may represent mechanical system wear over time (e.g., engineperformance degradation as noted by increased mechanical energy neededto produce the same work) or operational anomaly (part failure). Theflexible statistical model can be applied to data collected over amechanical system's life, to account for irrelevant factors and generatea predictive model of operational states such as wear, degradation oranomalies. Using the model adjusts for environmental and other factorsand references back to a consistent wear point. By example, for aircraftengines, the method may use the statistical model to model aircraft gasturbine engine exhaust gas temperature (EGT) as a function ofenvironmental, flight and engine parameters and to generate the outputdata as a predictive model of engine wear.

Still referring to FIG. 1, the method disclosed in U.S. PatentApplication Publ. No. 2010/0082267 further comprises the followingsteps: plotting the estimated operational states or wear chart (step132), using the plotted operational states or wear for trend analysis(step 134), and using the regression (random forest) methodology ormodel also, as a preliminary step 136 before detection and prognosisalgorithms, to make the mechanical system wear or degradation moreevident.

Improved Data-Driven Approach

The present invention improves upon the methodology described above. Thegeneral outline of this improved methodology is shown in FIG. 2. Themethod in accordance with one embodiment comprises: collecting data onthe aircraft engine via a data collection component (step 12);preprocessing the collected engine data to summarize operation of theengine (step 14); selecting a training data set that represents a basecondition for statistical comparison (step 16); fitting a statisticalempirical regression model to the training data set to model EGT as afunction of other variables (step 18), the result being an output model20; using the output model to produce residuals that can be interpretedas engine wear plus error (step 22); and using a time evolution model toestimate the operational state (e.g., wear) of the engine (step 24). Theestimated operational state 26 of the engine is then monitored (step28). A determination is then made whether the output of the monitorindicates an alert state (step 30). In response to a monitor outputindicating an alert state, an alert signal is issued and appropriateaction is taken in step 46. In the absence of a monitor outputindicating an alert state, the algorithm returns to step 22.

Data may be collected many times per second over flight. As discussedabove, for the purposes of trending EGT over the life of an engine, wefound it sufficient to choose a representative data point for eachflight. Our approach is to obtain the most stable engine operationalpoint possible, ideally at or near the maximum engine operating point.

Assume that an observed EGT measurement at a time t is a function ofother “outside” variables X_(t) (e.g., environmental, flight, and engineparameters):EGT _(t) =f(X _(t))+W _(t)+ε_(t)  (1)where ε_(t) is an error term consisting of measurement error, othervariables that have not been measured, and possible modelmisspecification; and W_(t) represents wear.

The previous work by Basu et al. estimated the function {circumflex over(f)}(X_(t)) using statistical regression. To estimate the function, theauthors experimented with linear, and various nonlinear parametricestimators. What worked the best is a random forest, which is a treeensemble method, as previously discussed above.

The residuals r_(t) equal wear plus error:r _(t) =EGT _(t) −{circumflex over (f)}(X _(t))=W _(t)+ε_(t)  (2)In the previous work by Basu et al., the residuals were used as theestimate of wear. However, the real goal is to estimate the wear W_(t).Since W_(t) is unobserved, an improved estimate can be achieved by usinga state space approach to estimate a time evolution model that relatesthe residuals over time. One particular time evolution model is aDynamic Linear Model (DLM). The DLM is disclosed by West, M. andHarrison, J. (1999). “Bayesian Forecasting and Dynamic Models”,Springer.

Wear over the life of a typical engine follows a roughly piecewiselinear pattern: a break-in period, followed by a relatively flatsection, followed by decreasing slope, possibly accelerating at the end.Therefore, a locally linear growth (or local linear trend, time varyingslope) model, also known as a second-order polynomial model, was used toestimate the wear W_(t).

The DLM that was used is defined by:

Observation Equation: given by Eq. (2).

State Evolution

Local Level:W _(t) =W _(t−1) +d _(t−1)+η_(t)  (3)Local Growth Rate:d _(t) =d _(t−1) +v _(t)  (4)The error terms η_(t) and v_(t) are assumed to be normally distributedwith mean 0 and covariance Σ. The wear at time t equals the wear W_(t−1)at the previous time plus a local growth rate d_(t−1). The local growthrate is a random walk. The goal is to estimate the state sequence W_(t).

One implementation of the foregoing uses several packages from the opensource statistical software R: randomForest (Liaw and Wiener (2002)) anddlm packages (see Petris, “dlm: Bayesian and Likelihood Analysis ofDynamic Linear Models. R package version 1.1-1 (2010)). Petris et al.(2009) describe the dlm package.

The methodology used in the previous effort by Basu et al. only requiresthe current data when predicting and generating residuals using a fittedrandom forest model. The random forest predicts each observationindependently of the past.

In the embodiment of the invention disclosed herein, the DLM relatesestimations over time. At each iteration, information about the past isencoded and saved as the parameters of the DLM from the previousiteration. These parameters need to be stored for each aircraft/engine.If DLM estimates from the previous iteration do not already exist, thenthe program gets default starting values, which represent an aggregatemodel estimated from historical engine data.

In one embodiment, default starting values may be generated by fitting aDLM model to the random forest residuals for each engine in the trainingdata. The default starting value for the observation variance is themedian. For the state estimates, the first 15 random forest residuals ofeach engine data series are used to estimate a linear regression. Theintercept and slope represent initial estimates for wear and growth forthat engine. The default starting value for the local level and growthrate are the medians.

In this way, one set of starting values is used for all engines. Anotherpossibility is to cluster engines, and using separate sets of startingvalues. Yet another possibility is to use a small set of data at thebeginning of engine life to individually estimate a DLM for each engine.This latter approach would mean that at the beginning of engine life,the program returns just the estimation from the random forest, withoutyet relating them in time.

In one embodiment, the evolution variances (i.e., the diagonal elementsof Σ) are set to large values. This allows the data to “speak forthemselves”, as is suitable when starting out. The evolution variance isestimated in subsequent iterations using a discount factor δ=0.9 (seePole et al. (1994)).

Monitor

In experiments, the RF+DLM model above showed improvement over both theOEM solution and the previous solution using just RF residuals. In orderto deploy this model in real life solutions, a working implementationshould also reject outliers and adapt model parameters after structuralchanges such as a sudden EGT shift.

The approach used here is based on the work by West (West, M., “BayesianModel Monitoring”, J. Royal Statistical Society, Series B, Vol. 48, pp.70-78 (1986)) and West and Harrison (West, M. and Harrison, J.,“Monitoring and Adaptation in Bayesian Forecasting Models”, J. AmericanStatistical Assoc., Vol. 81, No. 395, pp. 741-750 (1986)). This approachis useful and practical in many situations, and has the advantage thatdiscount factors (described below) allow closed form calculations.

Sequential Bayesian modeling analyzes observations in real time,updating inferences and predictive statements using newly obtainedinformation and observations. It assesses model fit using predictivedistributions. The idea behind monitoring is to compare the predictivefit of the standard model with an alternative model that specifies thenature of “unusual”. The central problem is to construct suitablealternatives to the “standard” model used for analysis. In this work,the alternative model is similar in form to the standard, but allows forchanges in the values of the parameters.

Monitoring is based on the Bayes factor, the ratio of likelihoods, whichcompares the predictive ability of the standard model versus thealternative model. It detects discrepancies between the data andstandard model predictions. Examples of model failures include outliersand structural changes in the time series. More formally, the Bayesfactor at time t is defined as:H _(t) =p(y _(t) |D _(t−1))/p _(A)(y _(t) |D _(t−1))  (5)Where a D_(t−1) is the data until time (t−1), and y_(t) is the currentobservation. Small values of H_(t) indicate poor performance of thestandard model relative to the alternative model. It is possible to puta threshold on (observation−forecast inconsistency): a Bayes factor>10gives evidence for the standard model, while >100 gives strong evidencefor the standard model; a Bayes factor<1/10 gives evidence for thealternative model, while <1/100 gives strong evidence for thealternative model.

The overall Bayes factor at time t is the product of the Bayes factorsuntil that time, and gives a measure of the global fit. However, theproblem with such a global measure is that the greater weight ofhistorical performance may mask local changes. For example, goodhistorical performance of the model may swamp a small Bayes factor dueto an outlier, which then goes undetected.

In a practical monitoring application, the local changes are of greatestinterest. A cumulative Bayes factor is a product of the most recent kBayes factors, and is sensitive to local model failure, and can indicateslow changes that may not be evident in a single Bayes factor.W _(t)(k)=H _(t) H _(t−1) H _(t−k+1) =H _(t) W _(t−1)(k−1)  (6)To focus on the most likely point of possible change, calculate the mostdiscrepant group of recent, consecutive observations as:V _(t)=min_(1≦k≦t) W _(t)(k)  (7)It turns out that when the cumulative evidence at time t favors thestandard model, so that V_(t−1)≧1, thenV _(t) =H _(t)  (8)and decisions about possible inadequacies are based on the currentobservation y_(t) alone. If the Bayes factor H_(t) is small enough, theny_(t) may be an outlier or the beginning of a structural change.

On the other hand, if the evidence before time t is against the standardmodel, so that V_(t−1)<1, then the cumulative Bayes factor is multipliedby H_(t):V _(t) =H _(t) V _(t−1)  (9)In this way, the monitor detects either gradual or abrupt changes.

When changes in parameter values are the primary cause of standard modelfailure, an additional goal is to automatically adapt to the onset ofchange. Incorporating increased uncertainties into the model leadsnaturally to more rapid adaptation by allowing future data to moreheavily influence the updating of posterior distributions. In this way,models self correct after structural changes. However, the automaticprocedure must also distinguish an outlier from a structural change.

The simple mode of operation of the model monitor using the sequences ofcumulative Bayes factors as described above provides an indication ofwhen outliers and changes may have occurred. When a change is signaledwith a group of consecutive observations identified as discrepant,action must be taken to adjust the model parameters to adapt to thechange. This is achieved in the disclosed embodiment by increasing theuncertainty in the prior distribution as measured by the priorcovariance matrix.

The following algorithm (disclosed by West and Harrison, 1986, citedabove) uses the monitor to isolate and reject outliers, or in cases ofstructural change, automatically increase uncertainty about theparameters to allow for rapid adaptation to new data in cases of change.

(A) If H_(t)≧τ, then y_(t) is consistent with the standard model. Butthe possibility of change before this time t should be assessed byproceeding to (B). If H_(t)<τ, then y_(t) may be an outlier that shouldbe rejected, or an indication of change, so proceed to (C).

(B) If the cumulative Bayes factor V_(t)≧τ, then proceed to (D) (updateas usual). Alternatively, if V_(t)<τ, then change is indicated; proceedto (C).

(C) Reject y_(t) as providing no useful information at time t about thestandard model parameters. Do not update model using y_(t)(equivalently, treat y_(t) as missing). Moreover, allow for change byincreasing the uncertainty about the parameter vector, leading to morerapid adaptation to new data. Increase the time index from t to t+1 andreturn to (A)

(D) Standard update: standard model is satisfactory. Update as usual tothe posterior and thence to the prior for time t+1. Return to (A).

FIG. 3 is a block flow diagram showing the Bayesian model monitoring andautomatic adaptation to model change being used in conjunction with thecombined (RF+DLM) model. Step 128 is the same as in FIG. 1, namely, theoutput model is used with preprocessed data to predict new data andcalculate residuals. In step 32, those residuals are related over timeusing the DLM, referred to in FIG. 3 as the TEM (i.e., time evolutionmodel). The resulting processed data 34, representing wear estimates, isthen inputted to a monitor, which detects either gradual or abruptchanges. More specifically, the monitor performs steps 36 and 42described hereinafter.

Still referring to FIG. 3, blocks 36, 38, 40, 42, 44 and 46 in FIG. 3implement the previously described algorithm for either updating the TEMto reflect gradual change or adapting to the TEM to allow for abruptchange. In step 36, a determination is made whether the data agrees withthe current TEM, i.e., whether the Bayes factor equals or exceeds thethreshold τ. If NO, then the time evolution model is adapted to allowfor abrupt change by increasing the uncertainty of the model (step 40)and the algorithm returns to step 32. If YES, then the cumulativeevidence is updated (step 38) and a determination is made in step 42whether the cumulative evidence agrees with the current time evolutionmodel, i.e., whether the cumulative Bayes factor equals or exceeds thethreshold τ. If NO, then the time evolution model is adapted to allowfor abrupt change by increasing the uncertainty of the model (step 40)and the algorithm returns to step 32. If YES, then the time evolutionmodel is updated (step 44) before returning to step 32.

To implement the foregoing algorithm, the state evolution covariancematrix must be specified. The values control the stochastic variation inthe evolution of the model, and determine the role of past observations.A key problem is that one covariance is typically not suitable for alltimes. Moreover, it is difficult to specify the covariance elements.

In the system equation, the covariance leads to an increase inuncertainty, or equivalently a loss of information, about the statevector between successive times. This idea is natural, and leads tospecifying the posterior covariance as a fraction 1/δ of the priorcovariance, and therefore the state evolution covariance as (1−δ)/δ ofthe prior covariance. The degree of adaptation to new data increases asthe discount factor becomes smaller.

In one embodiment, the combined (RF+DLM) model uses a discount factorδ=0.9. In the monitor, there are two other uses of the discount factor.First, the alternative model uses a discount factor of 0.05. Second, themonitor enables the combined model to adapt after a structural change bydecreasing the discount factor to 0.1.

A preliminary evaluation first looked at the monitor alerts and thedetected outliers for the same data set. The monitor generally alertswhen the slope changes direction or there is a shift. The authors ran asimple experiment to determine what magnitude shift the monitor candetect, and how long the detection takes. A shift was introduced byincrementing some of the data by 10, and 20 degrees. The threshold τ onthe Bayes factors affects the size of the shift that the algorithm candetect. For τ=0.2, the monitor detected a shift of 20 degrees, but not a15 degree shift. It took 15 observations to detect the 20 degree shift.But for a threshold τ=0.3, the monitor detected a shift of 15 degreesafter 25 observations. Deciding on the appropriate threshold depends onwhat magnitude shifts the user wants to detect, and also the sensitivityto other changes. Smaller threshold values decrease sensitivity toslower, less marked changes.

FIGS. 4A and 4B show a system in accordance with one embodiment of theinvention. As shown in FIG. 4A, a mechanical system 50 (e.g., a gasturbine engine) installed on a vehicle (e.g., an airplane) is monitored(in the manner described above) by a local health monitoring computer 52installed on the mechanical system. The health monitoring computer 52stores an empirical model 54 provided by a computer installed at amaster data station 56 (e.g., a ground station), shown in FIG. 4B. Theempirical model 54 can be either transmitted directly or by an antenna(not shown in FIG. 4B) at the master data station 56 to an antenna (notshown in FIG. 4A) onboard the airplane or any other suitable wirelesslink (indicated by dashed lines in FIGS. 4A and 4B).

Referring to FIG. 4B, block 58 represents a historical database ofoperational and environmental parameters (non-segregated) for themechanical system (item 50 in FIG. 4A). The computer at the master datastation 56 preprocesses the historical data in the manner previouslydescribed to create an initial segregated data set (step 60 in FIG. 4B)per data set segregation criteria for a time period. The computer at themaster data station 56 then creates a training data set (step 62) basedon the segregated initial historical data only, as previously described.That same computer then builds a statistical empirical regression model(step 64), which in the case of a gas turbine aircraft engine, modelsthe relationship between EGT and outside variables. The resultingempirical model 54 is then transmitted to the health monitoring computer52.

Referring to FIG. 4A, after the empirical model 54 has been loaded, thehealth monitoring computer 52 is ready to monitor the operation ofmechanical system 50. During operation of the mechanical system 50,various sensors (not shown) acquire current operational andenvironmental parametric data 66, which data is sent to the healthmonitoring computer 52. The health monitoring computer 52 is programmedwith software for preprocessing the current data to create a segregateddata set per data set segregation criteria for a time period (step 68).The health monitoring computer then calculates residuals using theempirical model 54. The health monitoring computer then uses a timeevolution model to predict the operational states of the mechanicalsystem (step 70). The resulting predicted state 72 is compared with theprocessed data and cumulative evidence in step 74. If the segregateddata matches the predicted state, then the time evolution model 70 isupdated (step 76). As previously explained with reference to FIG. 3, ifthe processed data and cumulative evidence do not agree with thepredicted state 76, the time evolution is adapted for extreme change(i.e., uncertainty of the model is increased) (not shown in FIG. 4A) andan event flag for corrective action is set (step 78 in FIG. 4A), whichevent flag setting is stored in computer memory. This flag can be sentvia radio immediately to inform maintenance or later when the airplanehas landed, a maintenance computer (not shown) can be connected to thehealth monitoring computer 52 and used to check the status of this eventflag. In response to receipt of a maintenance notification that theevent flag has been set to a state indicating that corrective action isrequired, maintenance personnel will attempt to isolate the fault in themechanical system which caused the disparity between the processed dataand the predicted state (step 80 in FIG. 4A).

While the invention has been described with reference to variousembodiments, it will be understood by those skilled in the art thatvarious changes may be made and equivalents may be substituted forelements thereof without departing from the scope of the invention. Inaddition, many modifications may be made to adapt a particular situationto the teachings of the invention without departing from the essentialscope thereof. Therefore it is intended that the invention not belimited to the particular embodiment disclosed as the best modecontemplated for carrying out this invention.

As used in the claims, the term “computer system” should be construedbroadly to encompass a system which has at least one computer orprocessor, and may have two or more computers or processors. Also, thesteps recited in the method claims should not be construed to requirethat such steps be performed in alphabetical order or in the order inwhich they are recited.

The invention claimed is:
 1. A method for monitoring wear in amechanical system, comprising: (a) repeatedly measuring a parameter overtime during operation of the mechanical system; (b) calculatingresiduals at time points over the life of the mechanical system using anempirical model that models values of said parameter as a function ofvalues of other parameters, said residuals representing the respectivedifferences between each measurement of said parameter and eachcorresponding parameter value predicted by the empirical model; (c)determining whether the measurements evolve as expected under a timeevolution model that relates predictions of residuals over time; and (d)flagging an event in response to the measurements of said parameterdeviating over time from the behavior predicted by the time evolutionmodel by more than a threshold value, wherein in accordance with saidtime evolution model, wear at time t equals the wear at a previous time(t−1) plus a local growth rate at time (t−1).
 2. The method as recitedin claim 1, further comprising isolating and rejecting outliers andadapting the time evolution model to reflect the onset of a structuralchange in response to the measurements deviating over time from thebehavior predicted by the time evolution model by more than saidthreshold value.
 3. The method as recited in claim 1, wherein saidadapting comprises increasing the uncertainty in a prior distribution asmeasured by a prior covariance matrix.
 4. The method as recited in claim1, further comprising updating the time evolution model in response tothe measurements deviating over time from the behavior predicted by thetime evolution model by less than said threshold value.
 5. The method asrecited in claim 4, wherein the local growth rate is a random walk. 6.The method as recited in claim 1, wherein said time evolution model is asecond-order polynomial dynamic linear model.
 7. The method as recitedin claim 1, wherein step (c) comprises comparing the respectivepredictive abilities of standard and alternative time evolution models,wherein said alternative model is similar in form to said standardmodel, but allows for more extreme wear observations.
 8. The method asrecited in claim 1, further comprising isolating a fault in saidmechanical system when said event is flagged.
 9. The method as recitedin claim 1, wherein said mechanical system is a gas turbine engine andsaid parameter is engine exhaust gas temperature.
 10. A method formonitoring wear in a mechanical system, comprising: (a) repeatedlymeasuring a parameter over time during operation of the mechanicalsystem; (b) calculating the value of a monitoring statistic for each ofsaid measurements; (c) calculating the value of a cumulative monitoringstatistic that is a product of sequential values of said monitoringstatistic; (d) determining whether the value of said monitoringstatistic is less than or not less than a threshold value; (e)determining whether the value of said cumulative monitoring statistic isless than or not less than said threshold value; and (f) flagging anevent in response to the values of said monitoring statistic and saidcumulative monitoring statistic being less than said threshold value,wherein said mechanical system is a gas turbine engine and saidparameter is engine exhaust gas temperature.
 11. The method as recitedin claim 10, wherein said event is an abrupt structural change in saidmechanical system.
 12. The method as recited in claim 10, furthercomprising isolating and rejecting outliers included in saidmeasurements and increasing the uncertainty in a prior distribution asmeasured by a prior covariance matrix to reflect the onset of astructural change in response to the values of said monitoring statisticand said cumulative monitoring statistic being less than said thresholdvalue.
 13. The method as recited in claim 10, further comprisingupdating a time evolution model in response to the values of saidmonitoring statistic and said cumulative monitoring statistic being notless than said threshold value.
 14. The method as recited in claim 10,further comprising isolating a fault in said mechanical system when saidevent is flagged.
 15. A system for monitoring the health of a mechanicalsystem, comprising a computer system programmed to perform the followingoperations: (a) receiving values representing the results ofmeasurements of a parameter over time during operation of the mechanicalsystem; (b) predicting residuals at time points over the life of themechanical system using an empirical model that models values of saidparameter as a function of values of other parameters, said residualsrepresenting the respective differences between each measurement of saidparameter and each corresponding parameter value predicted by theempirical model; (c) determining whether the measurements evolve asexpected under a time evolution model that relates predictions ofresiduals over time; and (d) flagging an event in response to themeasurements of said parameter deviating over time from the behaviorpredicted by the time evolution model by more than a threshold value,wherein said mechanical system is a gas turbine engine and saidparameter is engine exhaust gas temperature.
 16. The system as recitedin claim 15, wherein said computer system is further programmed toperform the following operations: isolating and rejecting outliers andadapting the time evolution model to reflect the onset of a structuralchange in response to the measurements deviating over time from thebehavior predicted by the time evolution model by more than saidthreshold value.
 17. The system as recited in claim 16, wherein saidadapting comprises increasing the uncertainty in a prior distribution asmeasured by a prior covariance matrix.
 18. The system as recited inclaim 15, wherein said computer system is further programmed to updatethe time evolution model in response to the measurements deviating overtime from the behavior predicted by the time evolution model by lessthan said threshold value.
 19. The system as recited in claim 15,wherein step (c) comprises comparing the respective predictive abilitiesof standard and alternative time evolution models, wherein saidalternative model is similar in form to said standard model, but allowsfor more extreme changes in the mechanical system or component.
 20. Asystem for monitoring the health of a mechanical system, comprising acomputer system programmed to perform the following operations: (a)receiving values representing the results of measurements of a parameterover time during operation of the mechanical system; (b) calculating thevalue of a monitoring statistic for each of said measurements; (c)calculating the value of a cumulative monitoring statistic that is aproduct of a plurality of values of said monitoring statistic; (d)determining whether the value of said monitoring statistic is less thanor not less than a threshold value; (e) determining whether the value ofsaid cumulative monitoring statistic is less than or not less than saidthreshold value; and (f) flagging an event in response to the values ofsaid monitoring statistic and said cumulative monitoring statistic beingless than said threshold value, wherein said mechanical system is a gasturbine engine and said parameter is engine exhaust gas temperature.